When should we use algorithms like Adam as opposed to SGD?
Empirically, I observed that algorithms like Adam and RMSProp tended to give me a final higher performance (in my case, the accuracy) on (the validation dataset) with respect to SGD. However, I also observed that Adam and RMSProp are highly sensitive to certain values of the learning rate (and, sometimes, other hyper-parameters like the batch size) and they can catastrophically fail to converge if e.g. the learning rate is too high. On the other hand, in general, SGD have not led me to the highest performance, but they did not catastrophically fail (at least, as much as Adam and RMSProp) in my experiments (even when using quite different hyper-parameters). I noticed that the learning rate (and the batch size) are the hyper-parameters that mainly affect the performance of all these algorithms.

In my experiments, I use SGD without momentum and I used the (PyTorch) default values of Adam and RMSProp. I only compared SGD with Adam and RMSProp, on the relatively simple task of recognising MNIST digits. You can have a look at this repository https://github.com/nbro/comparative-study-between-optimizers, which contains the code I used to perform these experiments. You also have the instructions there to perform the experiments (if you want).